Val: Difference between revisions
This is just unencyclopedic writing |
Not sure why giving a "bad" example in the first place so that you have to include the twist of logic. Amend it by plainly stating what we mind and what we don't. Cleanup +1 |
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If you want to find an approximation to a just interval, the immediate question is: why would you need an algorithm instead of just looking at the [[direct approximation]] possible in the edo? The answer is to avoid contradictions. For example, it might not be true that [[~]][[6/5]] × [[~]][[5/4]] = [[~]][[3/2]] or that [[~]]9/1 × [[~]]5/1 = [[~]]45/1 if you are just always using the direct approximation of each of these frequency ratios (6:5, 5:4, 3:2, 9:1, etc.) in the edo, because of something called ''inconsistency'', which means if you know what the intervals that you want to combine are, then combining their approximations in the edo does not give you the same result as multiplying their ratios ''first'' and ''then'' using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]]. | If you want to find an approximation to a just interval, the immediate question is: why would you need an algorithm instead of just looking at the [[direct approximation]] possible in the edo? The answer is to avoid contradictions. For example, it might not be true that [[~]][[6/5]] × [[~]][[5/4]] = [[~]][[3/2]] or that [[~]]9/1 × [[~]]5/1 = [[~]]45/1 if you are just always using the direct approximation of each of these frequency ratios (6:5, 5:4, 3:2, 9:1, etc.) in the edo, because of something called ''inconsistency'', which means if you know what the intervals that you want to combine are, then combining their approximations in the edo does not give you the same result as multiplying their ratios ''first'' and ''then'' using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]]. | ||
Unfortunately, when this happens, it is not possible to fix the inconsistency, except by using a different edo that approximates the intervals better so that the multiplication or division of their approximations is consistent, but that is not actually necessary. Plus, even if you did that, there would still be other inconsistent ratios because an approximation cannot be perfect, so you cannot truly eliminate the inconsistency completely. Rather than giving up and saying that we cannot guarantee that ~6/5 × ~5/4 = ~3/2 or ~9/1 × ~5/1 = ~45/1 or ~135/128 × ~24/25 = ~81/80, etc. in our chosen edo, it turns out we ''can'' actually guarantee this if we are willing to allow one or more of these ratios to | Unfortunately, when this happens, it is not possible to fix the inconsistency, except by using a different edo that approximates the intervals better so that the multiplication or division of their approximations is consistent, but that is not actually necessary. Plus, even if you did that, there would still be other inconsistent ratios because an approximation cannot be perfect, so you cannot truly eliminate the inconsistency completely. Rather than giving up and saying that we cannot guarantee that ~6/5 × ~5/4 = ~3/2 or ~9/1 × ~5/1 = ~45/1 or ~135/128 × ~24/25 = ~81/80, etc. in our chosen edo, it turns out we ''can'' actually guarantee this if we are willing to allow one or more of these ratios to ''not'' use the closest approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will allow us to do that, which brings us to… | ||
== Definition == | == Definition == | ||
A [[val]] is a list of numbers telling you the approximation of each [[prime harmonic]] used in an edo in terms of steps, where by ''prime harmonic'' we mean each frequency ratio ''p''/1 (where ''p'' is a {{w|prime number}}. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list | A [[val]] is a list of numbers telling you the approximation of each [[prime harmonic]] used in an edo in terms of steps, where by ''prime harmonic'' we mean each frequency ratio ''p''/1 (where ''p'' is a {{w|prime number}}. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list. 1st place is prime 2's mapping (a.k.a. the edo), 2nd place is prime 3's mapping, 3rd place is prime 5's mapping, 4th is prime 7's, etc. The val is used to understand the edo's approximations to ratios involving those primes, like 2 × 5 / 3 / 3 = [[10/9]] for primes {2, 3, 5}. This list [[#Warts and generalized patent vals|does not have to be the closest approximation]] for each prime, but it usually is. Thus a val is essentially just a list of numbers that we are interpreting as having a certain meaning. | ||
=== Example: 26edo === | === Example: 26edo === | ||